ENLARGEMENT OF FILTRATION AND THE STRICT LOCAL MARTINGALE PROPERTY IN STOCHASTIC DIFFERENTIAL EQUATIONS Aditi Dandapani Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the Graduate School of Arts and Sciences COLUMBIA UNIVERSITY 2016 © 2016 Aditi Dandapani All Rights Reserved Abstract ENLARGEMENT OF FILTRATION AND THE STRICT LOCAL MARTINGALE PROPERTY IN STOCHASTIC DIFFERENTIAL EQUATIONS Aditi Dandapani In this thesis, we study the strict local martingale property of solutions of various types of stochastic differential equations and the effect of an initial expansion of the filtration on this property. For the models we consider, we either use existing criteria or, in the case where the stochastic differential equation has jumps, develop new criteria that can can detect the presence of the strict local martingale property. We develop deterministic sufficient conditions on the drift and diffusion coefficient of the stochastic process such that an enlargement by initial expansion of the filtration can produce a strict local martingale from a true martingale. We also develop a way of characterizing the martingale property in stochastic volatility models where the local martingale has a general diffusion coefficient, of the form µ(St; vt);where the function µ(x1; x2) is locally Lipschitz in x: Contents Acknowledgements ii Introduction 1 Chapter 1 5 0.1 The One-Dimensional Case . .5 0.2 Stochastic Volatility Models . 13 0.3 Expansion of the Filtration by Initial Expansions . 31 Chapter 2 35 0.4 The Model of Lions and Musiela . 35 0.5 The Case of Jump Discontinuities . 62 0.6 The Model of Mijatovic and Urusov . 73 0.7 Examples . 81 Chapter 3 88 0.8 Stochastic Stability . 88 i 0.9 The Martingale Property in the Case of a General Diffusion Coefficient 95 0.10 Explosion of Multidimensional Diffusions . 102 0.11 An Application of Exponential p-Stability . 110 Conclusion 114 Bibliography 118 ii Acknowledgements I would first like to thank my parents, Vijay and Kamini Dandapani, for their unwavering support during my life and, specifically, the journey that has been my PhD. Without them, none of this would have been remotely possible. I would like to thank my grandparents, Paspa, Thath and Paku for their constant infusions of love and encouragement. I would like to mention and thank my grandmother, Pasma; though you are no longer here with me, I will never forget how you nurtured and adored me as a child. I would like to thank my brother, Rohan Dandapani for his unique way of showing support via comic relief, which enabled me to laugh and maintain my composure during crucial moments. I would also like to thank my uncle and aunt, Sekar and Ambika and my cousin Vishwanath for the love and support they have given me since childhood. I don't know how I can sufficiently express my gratitude to Professor Philip Protter for having taught, guided, helped, nurtured and supported me throughout my expe- rience as a doctoral researcher. I feel incredibly fortunate to have benefited from his iii tutelage during this time. He has been an excellent personal and academic mentor, and I have blossomed more than I could have hoped for both as a person and as a mathematician thanks to him. I would also like to thank Diane Rubenstein for her delightful company and her interest in my well-being. I would like to thank Professor Polvani for his support and encouragement during good and bad times. I would like to thank Christina Rohm, Montserrat Fernandez-Pinkley and Dina Amin for their support and help. I would also like to thank Professor Richard Davis, Dood Kalicharan and Anthony Cruz for their support and help and for welcoming me so warmly to their department and treating me like one of their own. I would like to thank the members of my committee, Professors Bal, Leung, Polvani, Tippett and Protter, for their time and interest in my work. I would like to thank Professors Paul Glasserman and Suresh Sundaresan for their very useful advice and discussions. I would like to thank Professor Nita Baxani for being a great teacher, lovely person, and a very positive influence on my life. I would like to thank Uttara Natarajan, Ravi Vaidhayanthan, Kartik Natarajan and Latha Dulipsingh for their encouragement, for providing lots of laughs and for all the very enjoyable times we have had throughout the years. I would like to mention that I have had the fortune of having my great great grand- iv father M.D. Narayana Iyer, my great grandmother N.Padmavathy and my grand- mother Vasanta Ramakrishnan as role models in my academic pursuits. Thank you to Professor Raghu Varadhan for some very useful and interesting con- versations. Thank you to my adorable darling dog Roger, whose unconditional affection and delightful presence has offered me much pleasure and comfort during happy and stressful times. Last, but not least, I would like to thank Anish King, Anna Kovacs, Antonio Frasca, Nandini Sur, Farah Al-Mansoor, Spencer Greenberg, Pablo Garcia, Prab- hakar Pundir, Ram Ramaswamy, Marco Santoli, Vanessa Mesics, Alexis Antoniades, Meena D'Souza, Vinay Satish, Marie Lesage, Omar El-Euch, Neela Zareen, Max- imilien Giacomin, Veronique Quefellec, Aditi Alaganan, Varsha Raghavan, Shuchi Vyas, Lisha Qiu, Romain Menegaux, David Ward, and Roseline Bilina-Falafala. You are all so beautiful, kind, and patient with me and you look after me so well. There is no way I could have done this without you. v Introduction In this thesis, we devote ourselves to studying mechanisms by which strict local martingales can arise from martingales. A strict local martingale is a local martin- gale which is not a martingale. We will study how expanding the original filtration with respect to which a process is a martingale can lead to a strict local martingale. That is, if we begin with a probability space (Ω; F; F;P ) where F denotes (Ft)t≥0, and with an F martingale M = (Mt)t≥0, and consider an expanded filtration G such that, for all t we have the inclusion Ft ⊂ Gt; when can we obtain a filtration G such that M becomes a strict local martingale, possibly under a different but equivalent probability measure Q? Strict local martingales have recently been a popular subject of study. Some rel- atively recent papers concerning strict local martingales include Biagini et al [3], Bilina-Protter [4], Chybiryakov [7], Cox-Hobson [8], Delbaen-Schachermayer [9], F¨ollmer-Protter[13], Lions-Musiela [29], Hulley [17], Keller-Ressel [24], Klebaner- Liptser [26], Kreher-Nikeghbali [27], Larsson [28], Madan-Yor [31], Mijatovic-Urusov [32], 1 Protter [37], Protter-Shimbo [38], and Sin [40], and from this list we can infer a certain interest. Our motivation comes from the analysis of financial bubbles, as explained in [37], for example. The theory tells us that on a compact time set, the (nonnegative) price process of a risky asset is in a bubble, i.e., undergoing specula- tive pricing, if and only if the price process is a strict local martingale under the risk neutral measure governing the situation. Therefore one can model the formation of bubbles by observing when the price process changes from being a martingale to being a strict local martingale. This is discussed in detail in [22], [3], and [37], for example. The models on which we perform an initial expansion of the filtration are stochastic volatility models. We work with the settings examined in Lions and Musiela [29], Mijatovic-Urusov [32] and the case in which the martingale driving the diffusion has jumps. In these cases, we assume always that a component of the stochastic volatility process is an It^odiffusion, so that we can use Feller's test for explosions in our quest to characterize the stochastic processes in question. An expansion of the filtration using initial expansion involves adding the informa- tion encoded in a random variable to the original σ algebra at time zero. This augmentation doesn't have to happen at time zero, however; it can happen at any stopping time. This random variable will be denoted L. We will see that this type of enlargement of filtration from F to G changes the risk-neutral measure from P to Q and our stochastic process, that we will call S and which is assumed to be a (P; F) 2 martingale, becomes a strict local martingale under (Q; G) on a stochastic interval that depends on the model and the random variable that we add to F: The case of initial expansions is particularly tractable, since Jacod [20] has devel- oped the theory that provides us with the dynamics of the process under the enlarged filtration. (See [36, Chapter VI] for a pedagogic exposition.) That is, he provides us with the semimartingale decomposition of the process in the enlarged filtration, which permits us to detect the presence of the strict local martingale property of the process, or lack thereof. In Chapter 1 we begin with an introduction to the paper of Delbaen and Shirakawa, which treats the one-dimensional case of a local martingale. After this, we re- introduce the models of P.L. Lions and M. Musiela, Mijatovic and Urusov, and Andersen and Piterbarg on stochastic volatility (in the style of what are known as Heston-type models). For each of these models, we conduct a detailed analysis of the techniques used to characterize the martingale property. In Chapter 2 we show how the addition of more information via an \expansion of the filtration” can lead what was originally a martingale to become a strict local martingale, under a risk neutral measure chosen from the infinite selection available in an incomplete mar- ket.